THE A2 THEOREM FOR PERFECT CALDERÓN-ZYGMUND
OPERATORS ON [0, 1)
POLONA DURCIK
Abstract. This is an outline of the proof of the A2 theorem by Lerner and Nazarov
[4], presented in the case of perfect Calderón-Zygmund operators on [0, 1). The approach of [4] is based on pointwise estimates of Calderón-Zygmund operators with
dyadic sparse operators, for which the theorem follows by a rather short argument.
We sketch this argument using the outer measure theory, referring to the recent work
by Thiele, Treil and Volberg [5].
1. Introduction
Let T be a perfect Calderón-Zygmund operator on [0, 1). That is, T is a bounded
operator on L2 ([0, 1)) and there is a kernel K, which is a locally integrable function on
{(x, y) ∈ [0, 1)2 : x 6= y}, such that
Z 1
T f (x) =
K(x, y)f (y)dy
0
whenever x ∈
/ supp(f ). The kernel K satisfies the size condition
C
|K(x, y)| ≤
|x − y|
for all x 6= y, and the smoothness condition
|K(x, y) − K(x0 , y)| = 0
(1)
x, x0
whenever
∈ I and y ∈ J for some disjoint dyadic intervals I and J. The symmetric
smoothness condition with x and y interchanged will not be needed. Moreover, we make
the technical assumption that K(·, y) vanishes on [1/2, 1).
By D = D([0, 1)) we denote the family of all dyadic intervals contained in [0, 1). Let
w be a weight, that is, an a.e. positive locally integrable function. A weight w ∈ A2 if
[w]A2 := sup(w)I (w−1 )I < ∞,
I∈D
where (w)I :=
1
|I|
R
I
w. The following is also known as the A2 theorem.
Theorem 1. For any w ∈ A2 , kT kL2 (w)→L2 (w) ≤ C(T )[w]A2 .
This bound on the norm of T in terms of the quantity [w]A2 holds not only in the
perfect case, but it is true for all Calderón-Zygmund operators on Rn . The bound is in
general sharp. There is a long history of proofs of the A2 theorem for different operators
T . In full generality it was first settled by Hytönen [1]. His and previous proofs for
special cases of T which were given by various authors all rely on representations of T in
terms of Haar shift operators. Recently, Lerner [2] gave an alternative proof based on estimating Calderón-Zygmund operators with dyadic sparse operators in the Banach space
norm. Lerner and Nazarov [4] went a step further by establishing pointwise domination
Date: November 28, 2014.
1
with dyadic sparse operators. We repeat their argument for perfect Calderón-Zygmund
operators on the unit interval, in which case it is significantly shorter.
2. Bounds for a function in terms of its λ−oscillation
This and the following section capture the main ideas from [4]. Our first goal is to
investigate bounds on functions in terms of their oscillation over sufficiently large sets.
Let f be measurable and finite a.e. on some measurable set I ⊂ R of finite positive
measure. For λ ∈ (0, 1) we define the λ−oscillation of f on I as
ωλ (f ; I) = inf{ω(f ; E) : E ⊂ I, |E| ≥ (1 − λ)|I|},
where ω(f ; E) := supE f − inf E f .
Proposition 2. Let f be a measurable function which is finite a.e. and supported on
[0, 1/2). For every ε > 0 there exists a family of dyadic intervals S ⊂ D satisfying
X
|J| ≤ 2|I| for all I ∈ D
(2)
J∈S,J⊂I
such that1
|f | ≤ ε +
X
ω 1 (f ; I)1I
a.e.
(3)
6
I∈S
Proof. We start with the construction of S. Let ε > 0. For each I ∈ D fix a set E(I) ⊂ I
with |E(I)| ≥ 56 |I| and
ω(f ; E(I)) ≤ ω 1 (f ; I) + 2−|I|−1 5−1 ε.
6
For an interval I ∈ D let I(I) be the family of all maximal intervals J ∈ D which are
contained in I and have the following property: there exists a child J 0 of J such that
E(J 0 ) ∩ E(I) = ∅.
The intervals in I(I) satisfy
X
J∈I(I)
1
|J| ≤ |I|.
2
(4)
Indeed, since E(J 0 ) ⊂ I \ E(I) and |J 0 | ≤ 65 |E(J 0 )|, we have
X
X
12 X
|J| = 2
|J 0 | ≤
|E(J 0 )|
5
J∈I(I)
J∈I(I)
J∈I(I)
12
1
≤ |I \ E(I)| ≤ |I|.
5
2
For the penultimate estimate we used disjointness of the sets E(J 0 ) ⊂ J, which follows
from the maximality of J ∈ I(I). The inequality (4) in particular implies I ∈
/ I(I), so
the maximality condition is well-defined.
Set I0 := {[0, 1)}. For every k ≥ 1 we inductively build Ik as
[
Ik :=
I(I).
I∈Ik−1
1By 1 we denote the characteristic function of an interval I.
I
2
We define the family S by
S :=
∞
[
Ik .
k=0
It satisfies the condition (2). To see that we first check the condition for the intervals
I ∈ S. Then I ∈ Ik0 for some k0 ≥ 0. By induction,
1 k−k0
X
|I|
|J| ≤
2
J⊂I,J∈Ik
for all k ≥ k0 . Summing over k ≥ k0 establishes the claim. Now let K ∈ D and
consider the family M of the maximal intervals I ∈ S which are contained in K. Using
disjointness of the latter we obtain
X
X X
X
|J| =
|J| ≤ 2
|I| ≤ 2|K|,
J∈S,J⊂K
I∈M J∈S,J⊂I
I∈M
as desired.
Now we show the bound (3). Consider I ∈ S. We ”go up” and enumerate all intervals
from S we meet on the way: I =: I1 ⊂ I2 ⊂ · · · ⊂ In := [0, 1). By the construction
of S, E(Ik ) ∩ E(Ik+1 ) 6= ∅ for all k = 1, . . . , n − 1. Take a point x ∈ E(I) and
y ∈ E(In−1 ) ∩ E(In ). Then
n−1
X
|f (x) − f (y)| ≤
ω(f ; E(Ij ))
j=1
and hence
|f (x)| ≤
n−1
X
ω(f ; E(Ij )) + sup |f |.
E(In )
j=1
By our assumption supp(f ) ⊂ [0, 1/2), so we have supE(In ) |f | ≤ ω(f ; E(In )). Thus
|f (x)| ≤
n
X
ω(f ; E(Ij )),
j=1
which can be restated as
|f | ≤ ε/5 +
X
on ∪I∈S E(I).
ω 1 (f ; I)1I
6
(5)
I∈S
It remains to extend the estimate (5) to a.e. on [0, 1). By the Lebesgue differentiation
theorem and Chebyshev’s inequality, for a.e. x ∈ [0, 1) we have that
|{y ∈ J : |f (x) − f (y)| > ε/5}| < |J|/6
(6)
for at least one interval J ∈ D containing x. Take any such x and consider an interval
J ∈ D containing x such that (6) holds. Let Je be the parent of J and denote by JeS the
e Since JeS ∈ S, we have
smallest interval in S containing J.
X
|f | ≤ ε/5 +
ω 1 (f ; I)1I
6
I∈S
on E(JeS ). By the construction of S, E(JeS ) ∩ E(J) 6= ∅. Thus,
X
ω 1 (f ; I)1I + ω 1 (f ; J)
|f | ≤ 2ε/5 +
6
I∈S
3
6
on E(J). We also have ω 1 (f ; J) ≤ ω(f ; Ax (J)) ≤ 2ε/5, where
6
Ax (J) := {y ∈ J : |f (x) − f (y)| ≤ ε/5}.
Since |Ax (J)| > 56 |J|, the sets E(J) ⊂ J and Ax (J) intersect. Choose y ∈ E(J) ∩ Ax (J).
Then we have
X
|f (x)| ≤ |f (y)| + |f (x) − f (y)| ≤ ε +
ω 1 (f ; I)1I (x).
6
I∈S
Thus,
|f | ≤ ε +
X
ω 1 (f ; I)1I
6
I∈S
almost everywhere on [0, 1).
3. Domination by dyadic sparse operators
In this section we apply Proposition 2 to establish pointwise domination of T f by
an operator associated with a family S, acting on |f |. In [4], such operators are called
sparse operators.
Theorem 3. For each f and every ε > 0 there exists S ⊂ D satisfying (2) such that
X
|T f | ≤ ε + C(T )
(|f |)I 1I a.e.
(7)
I∈S
The constant C(T ) does not depend on f .
Proof. Let I ∈ D. We start by estimating the difference of T f at two points x, x0 ∈ I.
By the triangle inequality we have
|T f (x) − T f (x0 )| ≤ |T (f 1I )(x) − T (f 1I )(x0 )| + |T (f 1I c )(x) − T (f 1I c )(x0 )|.
(8)
The second term vanishes, which can be seen from the integral representation of T
and the kernel condition (1). Indeed, if y ∈ I c , it is contained in a dyadic interval J ⊂ D
with I ∩ J = ∅, so
Z
0
|T (f 1I c )(x) − T (f 1I c )(x )| ≤
|K(x, y) − K(x0 , y)||f (y)|dy = 0.
Ic
The first term of (8) can be estimated using the weak (1, 1) boundedness of T . Namely,
for every α > 0, the set
Gα := {x ∈ I : |T (f 1I )(x)| > α}
satisfies the bound
|Gα | ≤ Cα−1 (|f |)I |I|.
Choose α := 6C(|f |)I . Then for every x, x0 ∈ E(I) := I \ Gα ,
|T (f 1I )(x) − T (f 1I )(x0 )| ≤ 2α = 12C(|f |)I .
Since |E(I)| ≥
(9)
5
6,
Proposition 2 applied to T f yields a family S ⊂ D such that
X
X
|T f | ≤ ε +
ω 1 (T f ; I)1I ≤ ε +
ω(T f ; E(I))1I a.e.
6
I∈S
I∈S
Together with (9) this establishes
|T f | ≤ ε + C(T )
X
(|f |)I 1I
a.e.
I∈S
as desired.
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4. The A2 theorem for sparse operators
To establish Theorem 1 it remains to prove it for sparse operators. We sketch the
argument using the outer measure theory of Do and Thiele [3], closely following the
setup in the work by Thiele, Treil and Volberg [5, Section 3 and Appendix].
Let us recall some definitions from [3] and [5]. The outer measure of an arbitrary
subset X ⊂ D is defined via coverings of X as
X
µ(X) := inf
|I|,
I
I∈I
S
where the infimum is taken over all I ⊂ D with X ⊂ I∈I D(I). For a function
F : D → C we define the following S 1 and S ∞ sizes as
1 X
|J||F (J)|
S 1 (F, D(I)) :=
|I|
J∈D(I)
and
S ∞ (F, D(I)) := sup |F (J)|,
J∈D(I)
respectively. For S being any of these two sizes we define the outer L∞ norm as
kF kL∞ (D,S) := sup S(F, D(I)).
I∈D
The outer
L1
norm is defined via the super level measure
µ(S(F ) > λ) :=
inf
X⊂D:kF 1D\X kL∞ (D,S) <λ
µ(X),
λ > 0, as
Z
∞
kF kL1 (D,S) =
µ(S(F ) > λ)dλ.
0
We have the following special case of the A2 theorem.
Proposition 4. Let a ∈ L∞ (D, S 1 ). Define the operator Aa by
X
Aa f =
a(I)(f )I 1I .
I∈D
Then for any w ∈ A2 , kAa kL2 (w)→L2 (w) ≤ 4kakL∞ (D,S 1 ) [w]A2 .
Proof. We may dualize and replace f by f w−1 , in order to deal with a more symmetric
situation. The inequality we need to establish then reads
X
|I||a(I)|(f w−1 )I (gw)I ≤ 4kakL∞ (D,S 1 ) [w]A2 kf kL2 (w−1 ) kgkL2 (w) ,
(10)
I∈D
for positive f and g, where the weighted L2 norm is defined as
Z 1
1/2
kf kL2 (w) :=
|f |2 w
.
0
We are set to use two tools from the outer measure theory, the Radon-Nikodym lemma
and the outer Hölder inequality. Both can be found in [3] or in [5, Appendix].
By the Radon-Nikodym lemma we can estimate the left hand-side of (10) by
ka(I)(f w−1 )I (gw)I kL1 (D,S 1 ) .
5
Using the outer Hölder inequality we can bound this by
−1
kakL∞ (D,S 1 ) k(w)I (w−1 )I kL∞ (D,S ∞ ) k(f w−1 )I (w−1 )−1
I (gw)I (w)I kL1 (D,S ∞ ) .
The second term in the last display is exactly the A2 constant [w]A2 . Hence it remains
to consider the third factor, which is taken care of by proving a bilinear embedding
theorem. We repeat the reasoning from [5, Theorem 3].
−1
Denote Fw−1 (I) := (f w−1 )I (w−1 )−1
I and Gw (I) := (gw)I (w)I . Our goal is to show
that the operator
f × g → Fw−1 Gw
2
−1
2
is bounded from L (w ) × L (w) to L1 (D, S ∞ ).
Let Iα be the collection of all maximal dyadic intervals I such that
Fw−1 (I)Gw (I) > α.
This implies that for any such I and any x, y ∈ I,
(Mwd −1 f )(x)(Mwd g)(y) ≥ α,
where
(f w)I
x∈I∈D (w)I
is the weighted Hardy-Littlewood dyadic maximal function. Thus, if I ∈ Iα ,
Mwd f (x) = sup
inf (Mwd −1 f · Mwd g) ≥ inf Mwd −1 f · inf Mwd g > α.
I
I
I
This implies
µ(S ∞ (Fw−1 Gw ) > α) ≤
X
|I| = |
I∈Iα
[
I| ≤ |{x ∈ [0, 1) : Mwd −1 f (x)Mwd g(x) > α}|.
I∈Iα
To finish the proof we integrate the last display over [0, ∞) in α. The right hand-side
becomes the L1 norm of Mwd −1 f · Mwd g, which is further estimated as
Z 1
Z 1
d
d
Mwd −1 f w−1/2 · Mwd gw1/2
Mw−1 f · Mw g =
0
0
Z
≤
0
1
(Mwd −1 f )2 w−1
1/2 Z
1
(Mwd g)2 w
1/2
≤ 4kf kL2 (w−1 ) kgkL2 (w) .
0
The last estimate follows from the general Hardy-Littlewood maximal theorem with
respect to the weights w and w−1 .
By Theorem 3, for each f there is S ⊂ D such that kT f kL2 (w) ≤ C(T )kA1S f kL2 (w) +
εkf kL2 (w) . Applying Proposition 4 to A1S , using k1S kL∞ (D,S 1 ) ≤ 2 and letting ε tend
to zero finishes the proof of Theorem 1.
References
[1] T. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math.
175 (2012), no. 3, 1473-1506.
[2] A. K. Lerner, A simple proof of the A2 conjecture, Int. Nath. Res. Not., 2013, no. 14, 3159-3170.
[3] Y. Do, C. Thiele, Lp theory for outer measures and two themes of Lennart Carleson united, to
appear in Bulletin AMS. arXiv:1309.0945
[4] A.
K.
Lerner
and
F.
Nazarov,
Intuitive
dyadic
calculus.
Available
at
http://www.math.kent.edu/ zvavitch/Lerner Nazarov Book.pdf
[5] C. Thiele, S. Treil, A. Volberg, Beyond the scope of doubling measures:
weighted
martingale transforms via outer measures, preprint. Available at http://www.math.unibonn.de/people/thiele/papers/OuterWittwer2014 10 15.pdf
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